Extending Stone duality to locally finite MV-algebras

ROBERTO CIGNOLI, EDUARDO J. DUBUC, DANIEL MUNDICI

JOURNAL OF PURE AND APPLIED ALGEBRA

Elsevier

Año: 2004 p. 37 - 59

0022-4049

Finite multisets are defined in combinatorics as pairs (X, s) where X is a finite set and s: X --> Natural Numbers is a map assigning a finite multiplicity to each element of X. MV-algebras were introduced by Chang as the algebraic counterpart of Lukasiewicz infinite-valued logic. They form a variety (or equational class) of algebras. Locally finite MV-algebras are filtered colimits of finite MV-algebras. In fact, the category of locally finite MV-algebras is the category of Ind-objects (in the sense of SGA4) over the category of finite MV-algebras. The main aim of this paper is to define a category C (whose objects we call multisets) such that the oposite category C^op is equivalent to the category LFMV of locally finite MV-algebras. The objects of C are pairs (X, s) where X is a Stone Space and s: X --> Supernatural Numbers is a continous map assigning a (possible infinite) multiplicity to each element of X. It will follow from the results in this paper that C is the category of pro-objects (in the sense of SGA4) over the category M of finite multisets. The main results of the paper is the equivalence of the categories C^op and LFMV. An important role in that proof is played by the category E of Hausdorff sheaves of rational MV-algebras over Stone spaces (or, inUniversal Algebra nomenclature, Boolean products of rational MV-algebras). As a matter of fact, we have that the categories E^op and LFMV, as well as the categories E and C are equivalent. Upon restriction to multisets with unit multiplicity our duality coincides with Stone´s duality between Stone spaces and Boolean algebras.